Investigation of the buckling of a parabolic
arch in a Vierendeel girder under compression
Autor(en):
Desprets, R.
Objekttyp:
Article
Zeitschrift:
IABSE congress report = Rapport du congrès AIPC = IVBH
Kongressbericht
Band (Jahr): 2 (1936)
PDF erstellt am:
16.06.2017
Persistenter Link: http://doi.org/10.5169/seals-3332
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V17
Investigation of the Buckling of a Parabolic Arch in
Vierendeel Girder under Compression.
a
Untersuchung über das Ausknicken des parabelförmigen
Druckgurtes eines VierendeekTrägers.
Etude du flambage d'ensemble de 1'arc parabolique
comprime d'une poutre Vierendeel.
R.
Professeur
ä
Desprets,
l'Universite de Bruxelles.
The parabolic arch of a Vierendeel girder, like that of a bowstring girder,
is subjeet to conditions as regards buckling which are difficult to define exactly,
for not only is it built in at its ends which form part of the lower (or tensile)
boom and of the roadway strueture, but at the same time it is rigidly fixed to
lhe verticais which are themselves very stiff and which connect v>ith the roadv>a\
strueture in the form of inverted portals.
It may nevertheless be assumed, at am rate in the central portion of the arch
wheiv the height of the verticais varies only slightly, that the arch acts like a
straight boom carried on verticais of constant height. By treating the arch
within this zone as if it were an ordinary straight boom, and by calculating the
load and the buckling length according to the well known methods of Engesser,
Timoshenko, Pigeaud, etc., it will thus be possible to arrive at a limiting stress.
With a view, however, to more exact understanding of the problem, attempts
have been made to apply Timoshenko's method taking due account of the end
fixations and assuming a shape of the deformed axis which agreeswell enough
with the truth but is also simple enough to avoid unduly complicating the cal¬
culations. Moreover, since the phenomenon of buckling will tend to make its
appearance in the central zone where the verticais are highest, it has been assumed
as a further simplification that the vertical members in question are of constant
height.
The numerical results show that the lengths of waves caculated in this way
justify the assumptions made. In studying ordinary buckling effects in a straight
piece, intermediately between joints, the deformed axis is compared to a simple
sinusoid; here, in order to take account of the conditions of end fixation, the
shape of the deformed axis was chosen in aecordance with the equation
y—
f (sin jti
K sin
3
jt y-j
640
V 17
R. Desprets
wherein the arch, supposed to be straightened along the median plane of the
beam, is taken as the x-axis; the origin is taken at one of the ends, and lhe
\-axis (parallel to which the deflections are measured) is taken at right angles to
the plane of the girder. It is further assumed that the thrust is constant troughout the arch between the two supports.
The principle underlying .the approximate method of Timoshenko is that of
equating the external work performed by the thrust with the work performed
by the ends of the verticais and by the internal stresses.
Work due to thrust.
The displacement of the thrust is equal to the difference in length between
the deformed arch and its projection:
f(ds —dx),
Ax=
(l +y'2)2dx
ds
o
where, approximately, ds
(1
ds
— dx
+ -^-y'2) dx
—- y'2
2 J
i,
f(ds — dx)
j,
Ax
U
TQ
Q
•
dx
Ax.
j
Py'2 dx
0
Resisting work performed by the ends of the uprights.
It will
be assumed that this thrust is continuous along the arch (Engesser).
be noted that this approximation has much better justification in the
should
It
Vierendeel girder than in the lattice girder, for in the former such continuity
is largely realised by the enlargement of the verticais at top and bottom, and the
assumption made is more likely to be true than that of isolated reactions. The
thrust at the end of a vertical is conditioned by two terms which depend on the
stiffness of the vertical member itself and on the stiffness of the cross girder
into which it frames.
If no account is taken of the reinforcing effect of the connecting gusset and h
is the height of the vertical member; p the span of the cross girder; lh and Ip
the moments of inertia of these respective members (e
EI), and Zf is the
deflection of the head of the vertical member under a load of one tonne, we
obtain
zf=h!lp+_hL
3
2
ep
eh
Moreover, the unit reaction continuously distributed is Cy where y is the
deformation at a given point and X is the width of the panel, whence
\CZf=l
and
C
-4r.
Investigation of the Bückling
of
Parabolic Arch in
a
a
Vierendeel Girder under Compression
641
The work done by the sides will be
l
y
-Jy2dx.
J JCydydx
o
o
Work due to internal stresses.
When account is taken only of the elastic energy of bending, we obtain the
expression
l
T
T,= CWA
2J7dx
0
or expressing M in terms of y',
JL
JL
0
I being
ü
assumed constant.
General equation.
As indicated above, the deformed shape of the neutral axis was taken to be in
aecordance with the equation
f (sin jty
y
where y
0 at the two ends and y
y' —
K sin 3 jt
f
(1
+
y)
K) at the centre
f y (cos ^ r- — 3 K cos 3 jt y-j
JV'dx
f*£.^(l+9K*)
0
L
J>dx
f*i±(l + K*)
0
y"
— fj2 (s*n n
JY'»dx
T
9 K sin 3 jt -y-j
p£.ii(i + 81K').
The fundamental relationship may be written
p-4-¥(1+9K,)==^(l+KÄ)+ifir»e(1 + 8lKt)
And finally:
U 1+K'
1+81K'
+
** 1+9K*"
V V+9K*
p_e^
41
V 17
642
For the
case
R. Desprets
of simple buckhng this takes the form
2
L2
X^+C^
P
first general term of Euler's formula and a complementary term
obtained from the ends of the vertical members. On determining the minimum
value of P in aecordance with the rule
dP
including
a
„
the following results are obtained, L being the buckling length:
The buckling load
P1
2^
L>1
2C-^;
JT
P^KC^;
U
nh; e^E-I.
To fix the minimum value for P according to the new hypothesis we may write
^.B.
P^.a + rL2
T2
L2
dP
The condition -j=-
0
n-
is expressed by
U 4 ~n_4
T
Li
^
e
B
C
»- Kb-'V^
+ 81 K2
1+9K2
1+K»
1
B
1
A
_
B—
1
+ 9K2
+ 81K2
1
+ K2
Juxtaposing these results with those above
L,
L,j/|
*
it
is seen that
Investigation of the Buckling of a Parabolic Arch in
a
Vierendeel Girder under Compression
643
To evaluate these expressions we must choose a value of K.
If the condition y' 0 is strictly observed at the ends we obtain
f£(l — 3K)
^-
K
0;
SJT2
This value of K will give the value 5 to the coefficient A in the term
—^
JU
for P. It appears that in the most favourable circumstances the maximum value
of A cannot exceed 4, which would correspond to perfect end fixation. This
— or K
would imply K2
we obtain the
—
approximately. On applying these values of K
following results:
(1)
W
(3)
'A3l
*K'-I9'
A
F2_LX2
^A^ + BJ-V
]ß
9
y'
B
3
1^
^-1/1 ^^3 ^.1.73
L,
L1-K2.45=L1-
1.565
7t2
IfK*
0,
L1
L2,
Pä=__2-E^.
Numerical application.
A numerical application of these results was made for the case of a parabolic
Vierendeel girder for a single track railway bridge of 100.10 m span and 14.30 m
rise (Vt*). divided into eleven panels of 9.10 m width.
Here Lx
Px
approximately 34 m.
2 X 1500
3000 tonnes.
Assuming a built-in arch with K2
If
Ls
Lj •
P2
t^- 2.85
K2
-_-,
9'
41
53 m
10
^i-y
L2
4250 t
L, __= Lt • 1.73
EJT2
P_,
1.565
t-
58.5 m
10
__
1500-y
Ls
5000t.
644
V 17
R. Desprets
For the arch under consideration the axial thrust due to füll live load amounts
to 1.035 tonnes at the crown and 1.230 tonnes at the springings, taking due
account of dynamic augment.
These calculations show that in the most unfavourable case, which is the
limiting condition for a girder without end fixation and with constant height,
the critical buckling load of the whole strueture represents a factor of safety of
nearly 3. Making the more correct assumption that the ends are to a considerable
extent built-in, and are firmly held by the ends of the vertical members and
cross girders, with K2
It
— the factor of safety will
be at least 4.
is found that the arch is of approximately the same strength whether con¬
sidered as bearing only on the verticais or as being fixed at the ends so that
its whole length acts as a single piece. It may also be concluded from these
results that the rigidity of the vertical members con fers upon the bridge a con¬
siderable margin of lateral strength; a margin which increases with the rigidity
of the verticais themselves and with the continuity of their connection to lhe
arch. These conditions are fulfilled best of all in the Vierendeel type of girder
here considered.